MEDICINE
REVIEW ARTICLE
Descriptive Statistics
The Specification of Statistical Measures and Their Presentation in Tables and Graphs
Part 7 of a Series on Evaluation of Scientific Publications
Albert Spriestersbach, Bernd Röhrig, Jean-Baptist du Prel, Aslihan Gerhold-Ay, Maria Blettner
SUMMARY
Background: Descriptive statistics are an essential part of
biometric analysis and a prerequisite for the understanding
of further statistical evaluations, including the drawing of
inferences. When data are well presented, it is usually
obvious whether the author has collected and evaluated
them correctly and in keeping with accepted practice in
the field.
Methods: Statistical variables in medicine may be of either
the metric (continuous, quantitative) or categorical
(nominal, ordinal) type. Easily understandable examples
are given. Basic techniques for the statistical description
of collected data are presented and illustrated with
examples.
Results: The goal of a scientific study must always be
clearly defined. The definition of the target value or clinical
endpoint determines the level of measurement of the
variables in question. Nearly all variables, whatever their
level of measurement, can be usefully presented
graphically and numerically. The level of measurement
determines what types of diagrams and statistical values
are appropriate. There are also different ways of presenting
combinations of two independent variables graphically and
numerically.
Conclusions: The description of collected data is
indispensable. If the data are of good quality, valid and
important conclusions can already be drawn when they are
properly described. Furthermore, data description provides
a basis for inferential statistics.
Key words: statistics, data analysis, biostatistics,
publication
Cite this as: Dtsch Arztebl Int 2009; 106(36): 578–83
DOI: 10.3238/arztebl.2009.0578
Institut für Medizinische Biometrie, Epidemiologie und Informatik, Johannes
Gutenberg-Universität Mainz: Prof. Dr. rer. nat. Blettner, Spriestersbach,
Gerhold-Ay
Zentrum Präventive Pädiatrie, Zentrum für Kinder- und Jugendmedizin, Universitätsmedizin der Johannes Gutenberg-Universität Mainz: Dr. med. du Prel,
M.P.H.
MDK Rheinland-Pfalz, Referat Rehabilitation/Biometrie, Alzey: Dr. rer. nat.
Röhrig
578
A
set of medical data is based on a collection of the
data of individual cases or objects, also called
observation units or statistical units. Every case, for
example every study participant, patient, every experimental animal, every tooth or every cell shows comparable parameters (such as body weight, gender, erosion,
pH). Each of these parameters, also called variables,
has a specific parameter value (gender = male, age =
30 years, weight = 70 kg) for each observation unit (for
example the patient). The aim of descriptive statistics
is to summarize the data, so that they can be clearly
illustrated (1–3).
The property of a parameter is specified by its socalled scale of measure. Generally two types of parameters are distinguished. A variable has a metric level
(= quantitative data) if it can be counted, measured or
weighed in a physical unit (as in cm or kg) or at least
can be recorded in whole numbers. Data with a metric
scale of measure can be further classified into continuous and discrete variables. In contrast to discrete
variables, continuous variables can take any value.
Examples for metric continuous parameters are body
height in cm, blood pressure in mmHg or the creatinine
concentration in mg/L. One example for a metric
discrete parameter is the number of erythrocytes per
microliter of blood.
The gender of man cannot be measured, but is classified into two categories. Parameters which can be
classified into two or more categories are described as
categorical parameters (= qualitative data). A further
classification of a categorical parameter is into nominal
characteristics (unordered) and ordinal characteristics
(ordered according to rank). Good basic portrayals of
the descriptive statistics of medical data can be found
in text books (4–9). Figure 1 gives a review of types of
parameters, as well as graphs to be used and statistical
measures.
Different procedures are necessary for the statistical
evaluation of metric and categorical parameters in graphic
and tabular forms. The graphs used here and the evaluation tables were created with the statistics package SPSS
for WINDOWS (Version 15). As example, we are using
data of 176 sportsmen and women.
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FIGURE 1
FIGURE 2
Example for a box plot
distribution of body weight in kg in 176 sportsmen and
women.
Sketch of parameter types and suitable statistical measures for descriptive presentation
1.1. Graphic presentation of a continuous parameter
1.1.2. The histogram
A histogram shows the distribution form of the measured
values of a continuous variable. In a normal distribution,
this takes the form of a "Gauss bell curve" (Figure 3a).
The present (measured) values are classified into an
appropriate number of classes (3). If the number of classes
is not "naturally" given, it is recommended to choose the
number of classes as the square root of the case number
N. If you had, for example, 49 cases, seven classes
would be chosen for the histogram on which the measured
values are distributed. Within each class, the measured
values are counted and illustrated as column in the figure.
Figure 3 shows five schematic examples for distribution
forms in each histogram.
In a histogram it is recognizable whether the data are
symmetrically distributed around the mean value (Figure
3a). However, when the histogram has a left peak (Figure
3b) or a right peak (Figure 3c), the values have a "skew"
distribution. In some situations, it may happen that
several peaks are recognizable in a histogram (Figure 3d
and e).
1.1.1. The box plot diagram
Box plots offer a visual impression of the position of the
first and third quartiles (25th and 75th percentile) and of
the median (central value). Also minimum, maximum,
and the breadth of scatter of all case values of a continuous parameter are recognizable. 50% of the values of
distribution are within the box (= interquartile range). A
box with a greater interquartile range indicates greater
scatter of the values. Figure 2 shows an example of the
1.2. Numerical description of a continuous parameter
The distribution of a recorded continuous parameter can
be numerically described with the following statistical
measures: Minimum, maximum, quartiles (with median),
range (difference between maximum and minimum),
skewness (indicates whether the distribution is symmetric
or not), arithmetic mean value, and standard deviation
(= square root of the variance) (6). When the parameter
"skewness" is between –1 and +1, the distribution is
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Results
1.
Description of a continuous parameter
A continuous parameter is a quantitative measure. The
value is measured on a continuous scale in arbitrarily
small intermediate steps (3). In this article, quantitative
parameters are treated as continuous parameters. The
size of the continuous parameters is presented by a measurement unit (= size unit, physical unit), for example
body height in cm or body weight in kg.
Initially, graphic presentations of the distribution of a
continuous variable are created in the form of box plots
and histograms. Such diagrams make it possible for the
researcher to get an initial visual impression of the distribution of the collected parameters.
MEDICINE
FIGURE 3
Examples of the types of distribution in histograms
a) Normal distribution (symmetric)
b) left peak (= skewed to the right)
c) right peak (= skewed to the left)
d) two peaks (symmetric)
e) several peaks
symmetric, but when it is under –1 or above +1 the values
are distributed with a right or a left peak (Box).
2.
Description of a categorical parameter
2.1. Graphic presentation of a categorical parameter
2.1.1. The pie diagram as graphic presentation of a categorical
parameter
The pie diagram or circular diagram is a popular form of
presentation for the distribution of characteristics of a
parameter classified into groups. The number of segments in one pie diagram corresponds to the number of
possible characteristics (= steps) of these variables. So,
a pie diagram would have two segments for "sex." Their
proportion in the total pie corresponds to their relative
percentage.
BOX
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2.1.2. The bar diagram as graphic presentation of a categorical
parameter
The bar diagram offers an alternative form of presentation. In contrast to the pie diagram, the frequency values
are read on the y-axis. This figure can present absolute
or relative frequencies. It is possible to compare the
heights of the bars directly, which is not possible with
the segments in the pie diagram. In contrast to the histogram for continuous parameters, there are no class intervals for the value ranges of the measurements on the
x-axis of the bar diagram. On the contrary, each bar
creates one unit completed to the right and to the left,
according to its value. Thus, one single bar refers either
to the female or to the male sex in Figure 4. In contrast
to the histogram, the single bars or columns of a bar diagram should contain gaps.
2.2. Numerical description of a categorical parameter
Both absolute and relative frequencies of a categorical
parameter can be given in a frequency table. In Table 1,
we show our population of sportswomen and sportsmen
once again: In this SPSS Version of a frequency table,
you can find the absolute frequencies in the "frequency"
column. If given, the number of missing values is listed
here. In the columns "percentage" and "valid percentages," you can choose whether the missing values
should be listed or not as a separate category. In the
column "cumulated percentages," the successive cumulated relative frequencies are presented. They are only
important in parameters of ordinal scale level with more
than two values. The two last columns ("valid and
cumulated percentages") are generally not suitable for
presentation in a publication.
3.
Description of correlations
Until now, we have only considered single variables, i.e.,
we have described our data as "univariate." No parameter
was related to any other. It is also possible to describe potential correlations between two variables, for example
between body weight (continuous) and body height (continuous). It is also possible to relate two categorical parameters or a metric and a categorical parameter.
3.1. Description of the correlation of two continuous
parameters
One of the variables is allocated to each of the two axes
in the scatter plot (point cloud diagram). If there is a correlation, it is expressed in the trend of the point cloud to
form an ellipse. If there were a 100 percent linear correlation between two parameters, all points would lie on a
straight line. In our example (10), the measured values
of bone density on the proximal site of measurement
(variable: 'spa_prox') and the values of bone density on
the distal site of measurement (variable: 'spa_dist') are
correlated with each other (Figure 5). If there were no
correlation, the area of the diagram would be unstructured
and covered with points.
The degree of correlation can be numerically expressed
as linear correlation with a coefficient of correlation.
This index can have values between –1 and +1. If both
distributions are symmetric, the coefficient of correlation
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MEDICINE
can be calculated according to Pearson. Otherwise, the
coefficient of correlation according to Spearman is
appropriate. This is not directly calculated from the
values but from their ranks. In linear distribution, the
coefficient of correlation should be calculated according
to Pearson. The Pearson coefficient of correlation is calculated as 0.886 for the measurement of bone density.
This is a very strong correlation.
3.2. Description of the correlation of two categorical
parameters
A grouped bar diagram is suitable for the graphical presentation of the correlation between two categorical
parameters. The standard form uses the x-axis for entry
of the single categories of the first variable and portrays
within these categories the absolute (number) or relative
frequencies (percentage) of the second variable through
different colored bars. In Figure 6, a possible correlation
between smokers and (smokers') cough is to be studied.
The bars represent relative frequencies in percentages.
They are more suitable for comparison than the absolute
frequencies, as the groups are often not of the same size.
In this figure, a bar diagram is presented and it is recognizable that smokers (with 71%) more often have cough
than non-smokers (29%).
The numerical description of information about absolute and relative frequencies of the two correlated
categorical variables is summarized in a cross table. We
differentiate between line variables (smokers and nonsmokers) and column variables (cough). It is possible to
give the absolute and relative frequencies, i.e. line and
column percentages, within the cells (Table 2).
You can see from this presentation that 21/29 = 72%
of non-smokers do not cough but 8/29 = 28% cough.
The cumulative percentage for all lines is 100% and the
cumulative percentage for all columns is also 100%.
The four field table helps the reader to understand the
reference values and the results.
To simplify the interpretation, it should be made clear
which variable is the target parameter. In our example, it
was important to know whether smoking influences the
occurrence of cough. Thus, cough is the (dependent) target parameter and smoking the (independent) possible
factor. In the interest of clarity of the cross table, it is
recommended to write the factor in the cells and the
target parameter in the columns. Then you can dispense
with the percentages in the columns. In this way, the
cross table is clearer and easier to understand. In practice, medical expertise is required to be able to formulate
a sensible question. This results in the arrangement of
the variables. Further parameters which can be calculated
from the four field table (relative risk, odds ratio) will be
discussed in a later publication.
3.3. Correlation between a continuous and a categorical
parameter
Several box plots can be presented in one diagram with
the help of a statistics program like SPSS, STATISTICA,
or SAS. Thus, it is possible to compare groups for which
for example the continuous parameter "weight in kg"
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Example of a bar
diagram of two
groups
FIGURE 4
TABLE 1
Sex
Valid
Male
Frequency
Percentage
102
58.0
Valid
Cumulated
percentages percentages
58.0
58.0
100.0
Female
74
42.0
42.0
Total
176
100.0
100.0
FIGURE 5
Example of a scatter diagram
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MEDICINE
Example of a
grouped bar
diagram
was recorded. In Figure 7, the distribution of the weight
of 74 sportswomen was compared with the distribution
of the weight of 102 sportsmen. When comparing the
two groups, a trend is recognizable: Men tend to have
higher weights than women (consideration of medians).
The values are more scattered with men than with
women, as the box plot for men is more stretched than
for women.
As described in chapter 1.2., suitable measures are
numerically calculated for the comparison of the groups
of the two sexes.
FIGURE 6
Discussion
TABLE 2
Distribution of the parameters smoking and cough
Cough
Smokers
No
Yes
Total
No
Number
% of smokers
% with cough
21
72%
75%
8
28%
29%
29
100%
52%
Yes
Number
% of smokers
% with cough
7
26%
25%
20
74%
71%
27
100%
48%
Number
% of smokers
% with cough
28
50%
100%
28
50%
100%
56
100%
100%
Total
Example of grouped
box plots
FIGURE 7
The exact description of data collected in a study is sensible and important. The correct descriptive presentation
of the results is the first step in evaluating and graphically presenting the results (7–9, 11). The description is
the basis of the biometric evaluation and is the indispensable starting point for further methodological procedures such as statistical significance tests. The
descriptive presentation of study results usually occupies
most of the space in publications. The description
covers the graphical and tabular presentation of the
results. The exact assessment of the scale level of the
parameter is important as the scale level determines the
type and procedure, both in the descriptive, and in the
explorative (= generating of hypothesis) and confirmatory (= biometric testing of hypothesis) evaluation. The
selection of a suitable statistical test procedure for controlling the significance is determined by the scale level
of the investigated parameters.
In normally distributed data, the arithmetic mean
value is the same as the median. The skewness has the
value zero. Unfortunately there rarely is a normal distribution in natural systems as in parameters collected in
patients. It is sensible to give the arithmetic mean value,
as well as the median for continuous data. A normal distribution cannot be assumed when the two values are
very different. The arithmetic mean value cannot be calculated in data with a purely ordinal scale. It is often
asked whether graphical or numerical presentations are
preferable in data description. Graphics serve to give a
first impression and to visually illustrate the situation of
distribution parameters. It can be difficult to read the
exact values of the median or the percentiles on the y-axis
in a box plot diagram. For this reason, calculation and
presentation of the exact statistical characteristic values
is indispensable.
In the individual case, the information of further biometric measures is of course valuable, including measures
not mentioned in the article. Examples would be effect
sizes, confidence intervals, Cohen's kappa, relative risk,
and cumulated values.
The use of suitable validated statistics software like
SPSS or SAS is recommended for statistical evaluation
of data.
Conflict of interest statement
The authors declare that there is not conflict of interest according to the
guidelines of the International Committee of Medical Journal Editors.
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Manuscript received on 4 February 2009, revised version accepted on
16 March 2009.
Translated from the original German by Rodney A. Yeates, M.A., Ph.D.
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Coresponding author
Prof. Dr. rer. nat. Maria Blettner
Institut für Medizinische Biometrie,
Epidemiologie und Informatik
Johannes Gutenberg-Universität
55101 Mainz, Germany
[email protected]
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